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In some cases, it is possible to solve such an equation for y as an explicit function (or several functions) of x.

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Nonetheless, x 3 + y 3 = 6xy is the equation of a curve called the folium of Descartes shown here and it implicitly defines y as several functions of x. It’s not easy to solve equation x 3 + y 3 = 6xy for y explicitly as a function of x by hand. A computer algebra system has no trouble. However, the expressions it obtains are very complicated Fortunately, we don’t need to solve an equation for y in terms of x to find the derivative of y. Instead, we can use the method of implicit differentiation. This consists of differentiating both sides of the equation with respect to x and then solving the resulting equation for y’. In the examples, it is always assumed that the given equation determines y implicitly as a differentiable function of x so that the method of implicit differentiation can be applied.

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Higher Order Derivatives Find if Substitute back into the equation.

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Find y” if x 4 + y 4 = 16. Differentiating the equation implicitly with respect to x, we get 4x 3 + 4y 3 y’ = 0. example: To find y’’, we differentiate this expression for y’ using the Quotient Rule and remembering that y is a function of x:

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However, the values of x and y must satisfy the original equation x 4 + y 4 = 16. So, the answer looks quite simple: